# Let $S(k)=1+3+5+...+(2k-1)=3+k^{2}$. Then which of the following is true? Option 1) $S(k)\Rightarrow S(k-1)\;$ Option 2) $\; \; S(k)\Rightarrow S(k+1)\; \; \;$ Option 3) $S(1)$  is correct Option 4) principle of mathematical induction can be used to prove the formula

As we learnt in

Steps of Mathematical Induction (Verification step) -

Step 1: Verification step

Actual verification of the proposition of the starting value $n=1$

- wherein

$2^{3n}-1$ is divisible by 7

Put n=1, It Satisfies.

and

Steps of Mathematical Induction (Induction Step) -

Step 2: Induction Step

Assuming the proposition to be true for n=k, and proving it is true for value     $n=k+1$

-

and

Steps of Mathematical Induction (Generalization Step) -

Combine Verification step and Induction step

$p(1)$ is true and $p(n)$ is true for $n+1$ assuming it is true for $n$

-

$S (k) = 1+3+5+------+ (2k -1)$

$= 3 + k^{2}$

$S (1) = 1^{1} = 3 + 1$ Which is not true

$\because S (1)$ is not true

$\therefore$ S (k) is true

$1+ 3+5+ -------- + (2k - 1) = 3 +k^{2}$

$S (k+1) = 1+3+5+-----+(2k - 1)+(2k + 1)$

$=3+k ^{2} + 2k +1$

$= 3 + (k + 1)^{2}$

$S \left ( k+1 \right ) \Rightarrow S\left ( k \right )$

Option 1)

$S(k)\Rightarrow S(k-1)\;$

This option is incorrect.

Option 2)

$\; \; S(k)\Rightarrow S(k+1)\; \; \;$

This option is correct.

Option 3)

$S(1)$  is correct

This option is incorrect.

Option 4)

principle of mathematical induction can be used to prove the formula

This option is incorrect.

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